Theorem ipasslem2 27687

Description: Lemma for ipassi 27696. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	|- X = ( BaseSet ` U )
ip1i.2	|- G = ( +v ` U )
ip1i.4	|- S = ( .sOLD ` U )
ip1i.7	|- P = ( .iOLD ` U )
ip1i.9	|- U e. CPreHil OLD
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem2	|- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( -u N x. ( A P B ) ) )

Proof of Theorem ipasslem2

Step	Hyp	Ref	Expression
1		nn0cn 11302	. . . . 5 |- ( N e. NN0 -> N e. CC )
2	1	negcld 10379	. . . 4 |- ( N e. NN0 -> -u N e. CC )
3		ip1i.9	. . . . . 6 |- U e. CPreHil OLD
4	3	phnvi 27671	. . . . 5 |- U e. NrmCVec
5		ipasslem1.b	. . . . 5 |- B e. X
6		ip1i.1	. . . . . 6 |- X = ( BaseSet ` U )
7		ip1i.7	. . . . . 6 |- P = ( .iOLD ` U )
8	6, 7	dipcl 27567	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
9	4, 5, 8	mp3an13 1415	. . . 4 |- ( A e. X -> ( A P B ) e. CC )
10		mulcl 10020	. . . 4 |- ( ( -u N e. CC /\ ( A P B ) e. CC ) -> ( -u N x. ( A P B ) ) e. CC )
11	2, 9, 10	syl2an 494	. . 3 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( A P B ) ) e. CC )
12		ip1i.4	. . . . . . 7 |- S = ( .sOLD ` U )
13	6, 12	nvscl 27481	. . . . . 6 |- ( ( U e. NrmCVec /\ -u N e. CC /\ A e. X ) -> ( -u N S A ) e. X )
14	4, 13	mp3an1 1411	. . . . 5 |- ( ( -u N e. CC /\ A e. X ) -> ( -u N S A ) e. X )
15	2, 14	sylan 488	. . . 4 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N S A ) e. X )
16	6, 7	dipcl 27567	. . . . 5 |- ( ( U e. NrmCVec /\ ( -u N S A ) e. X /\ B e. X ) -> ( ( -u N S A ) P B ) e. CC )
17	4, 5, 16	mp3an13 1415	. . . 4 |- ( ( -u N S A ) e. X -> ( ( -u N S A ) P B ) e. CC )
18	15, 17	syl 17	. . 3 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) e. CC )
19		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
20		mulneg2 10467	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ 1 e. CC ) -> ( N x. -u 1 ) = -u ( N x. 1 ) )
21	19, 20	mpan2 707	. . . . . . . . . . . 12 |- ( N e. CC -> ( N x. -u 1 ) = -u ( N x. 1 ) )
22		mulid1 10037	. . . . . . . . . . . . 13 |- ( N e. CC -> ( N x. 1 ) = N )
23	22	negeqd 10275	. . . . . . . . . . . 12 |- ( N e. CC -> -u ( N x. 1 ) = -u N )
24	21, 23	eqtr2d 2657	. . . . . . . . . . 11 |- ( N e. CC -> -u N = ( N x. -u 1 ) )
25	24	adantr 481	. . . . . . . . . 10 |- ( ( N e. CC /\ A e. X ) -> -u N = ( N x. -u 1 ) )
26	25	oveq1d 6665	. . . . . . . . 9 |- ( ( N e. CC /\ A e. X ) -> ( -u N S A ) = ( ( N x. -u 1 ) S A ) )
27		neg1cn 11124	. . . . . . . . . 10 |- -u 1 e. CC
28	6, 12	nvsass 27483	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ ( N e. CC /\ -u 1 e. CC /\ A e. X ) ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) )
29	4, 28	mpan 706	. . . . . . . . . 10 |- ( ( N e. CC /\ -u 1 e. CC /\ A e. X ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) )
30	27, 29	mp3an2 1412	. . . . . . . . 9 |- ( ( N e. CC /\ A e. X ) -> ( ( N x. -u 1 ) S A ) = ( N S ( -u 1 S A ) ) )
31	26, 30	eqtrd 2656	. . . . . . . 8 |- ( ( N e. CC /\ A e. X ) -> ( -u N S A ) = ( N S ( -u 1 S A ) ) )
32	1, 31	sylan 488	. . . . . . 7 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N S A ) = ( N S ( -u 1 S A ) ) )
33	32	oveq1d 6665	. . . . . 6 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( ( N S ( -u 1 S A ) ) P B ) )
34	6, 12	nvscl 27481	. . . . . . . 8 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ A e. X ) -> ( -u 1 S A ) e. X )
35	4, 27, 34	mp3an12 1414	. . . . . . 7 |- ( A e. X -> ( -u 1 S A ) e. X )
36		ip1i.2	. . . . . . . 8 |- G = ( +v ` U )
37	6, 36, 12, 7, 3, 5	ipasslem1 27686	. . . . . . 7 |- ( ( N e. NN0 /\ ( -u 1 S A ) e. X ) -> ( ( N S ( -u 1 S A ) ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) )
38	35, 37	sylan2 491	. . . . . 6 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S ( -u 1 S A ) ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) )
39	33, 38	eqtrd 2656	. . . . 5 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( N x. ( ( -u 1 S A ) P B ) ) )
40	39	oveq2d 6666	. . . 4 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) - ( ( -u N S A ) P B ) ) = ( ( -u N x. ( A P B ) ) - ( N x. ( ( -u 1 S A ) P B ) ) ) )
41	6, 7	dipcl 27567	. . . . . . . 8 |- ( ( U e. NrmCVec /\ ( -u 1 S A ) e. X /\ B e. X ) -> ( ( -u 1 S A ) P B ) e. CC )
42	4, 5, 41	mp3an13 1415	. . . . . . 7 |- ( ( -u 1 S A ) e. X -> ( ( -u 1 S A ) P B ) e. CC )
43	35, 42	syl 17	. . . . . 6 |- ( A e. X -> ( ( -u 1 S A ) P B ) e. CC )
44		mulcl 10020	. . . . . 6 |- ( ( N e. CC /\ ( ( -u 1 S A ) P B ) e. CC ) -> ( N x. ( ( -u 1 S A ) P B ) ) e. CC )
45	1, 43, 44	syl2an 494	. . . . 5 |- ( ( N e. NN0 /\ A e. X ) -> ( N x. ( ( -u 1 S A ) P B ) ) e. CC )
46	11, 45	negsubd 10398	. . . 4 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) - ( N x. ( ( -u 1 S A ) P B ) ) ) )
47		mulneg1 10466	. . . . . . 7 |- ( ( N e. CC /\ ( ( -u 1 S A ) P B ) e. CC ) -> ( -u N x. ( ( -u 1 S A ) P B ) ) = -u ( N x. ( ( -u 1 S A ) P B ) ) )
48	1, 43, 47	syl2an 494	. . . . . 6 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( -u 1 S A ) P B ) ) = -u ( N x. ( ( -u 1 S A ) P B ) ) )
49	48	oveq2d 6666	. . . . 5 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) )
50	2	adantr 481	. . . . . . 7 |- ( ( N e. NN0 /\ A e. X ) -> -u N e. CC )
51	9	adantl 482	. . . . . . 7 |- ( ( N e. NN0 /\ A e. X ) -> ( A P B ) e. CC )
52	43	adantl 482	. . . . . . 7 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u 1 S A ) P B ) e. CC )
53	50, 51, 52	adddid 10064	. . . . . 6 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) )
54	6, 36, 12, 7, 3	ipdiri 27685	. . . . . . . . . . 11 |- ( ( A e. X /\ ( -u 1 S A ) e. X /\ B e. X ) -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) )
55	5, 54	mp3an3 1413	. . . . . . . . . 10 |- ( ( A e. X /\ ( -u 1 S A ) e. X ) -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) )
56	35, 55	mpdan 702	. . . . . . . . 9 |- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( A P B ) + ( ( -u 1 S A ) P B ) ) )
57		eqid 2622	. . . . . . . . . . . . 13 |- ( 0vec ` U ) = ( 0vec ` U )
58	6, 36, 12, 57	nvrinv 27506	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ A e. X ) -> ( A G ( -u 1 S A ) ) = ( 0vec ` U ) )
59	4, 58	mpan 706	. . . . . . . . . . 11 |- ( A e. X -> ( A G ( -u 1 S A ) ) = ( 0vec ` U ) )
60	59	oveq1d 6665	. . . . . . . . . 10 |- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = ( ( 0vec ` U ) P B ) )
61	6, 57, 7	dip0l 27573	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
62	4, 5, 61	mp2an 708	. . . . . . . . . 10 |- ( ( 0vec ` U ) P B ) = 0
63	60, 62	syl6eq 2672	. . . . . . . . 9 |- ( A e. X -> ( ( A G ( -u 1 S A ) ) P B ) = 0 )
64	56, 63	eqtr3d 2658	. . . . . . . 8 |- ( A e. X -> ( ( A P B ) + ( ( -u 1 S A ) P B ) ) = 0 )
65	64	oveq2d 6666	. . . . . . 7 |- ( A e. X -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = ( -u N x. 0 ) )
66	2	mul01d 10235	. . . . . . 7 |- ( N e. NN0 -> ( -u N x. 0 ) = 0 )
67	65, 66	sylan9eqr 2678	. . . . . 6 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( ( A P B ) + ( ( -u 1 S A ) P B ) ) ) = 0 )
68	53, 67	eqtr3d 2658	. . . . 5 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + ( -u N x. ( ( -u 1 S A ) P B ) ) ) = 0 )
69	49, 68	eqtr3d 2658	. . . 4 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) + -u ( N x. ( ( -u 1 S A ) P B ) ) ) = 0 )
70	40, 46, 69	3eqtr2d 2662	. . 3 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N x. ( A P B ) ) - ( ( -u N S A ) P B ) ) = 0 )
71	11, 18, 70	subeq0d 10400	. 2 |- ( ( N e. NN0 /\ A e. X ) -> ( -u N x. ( A P B ) ) = ( ( -u N S A ) P B ) )
72	71	eqcomd 2628	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( -u N S A ) P B ) = ( -u N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NN0cn0 11292   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443   .iOLDcdip 27555   CPreHil OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-dip 27556  df-ph 27668

This theorem is referenced by:  ipasslem3  27688